7. A survey of $2 \times 2$ cases

So far we have not discussed whether all $2 \times 2$ matrices are like the three examples--and they aren't! On the other hand, not many different kinds of things can happen. Here are the typical cases:

1. Diagonal matrices. There are two subcases:
   1. Scalar matrices $r I = \matp{cc}{r&0\\ 0&r}$. Here the only eigenvalue is $\lambda = r$ and every nonzero vector is an eigenvector.

   2. Diagonal matrices with distinct diagonal entries $\matp{rr}{d&0\\ 0&e}$ with $d \neq e$. Here the eigenvectors lie on the $x$- and $y$-axes and the eigenvalues are just the diagonal entries.

2. Symmetric matrices, other than scalar matrices. In this case, there are two real eigenvalues and the eigenspaces (eigenvector directions) are orthogonal. See for instance the example in Figure .

3. Matrices with two distinct real eigenvalues, as in the example in Figure . Here there are two eigenspaces (lines of eigenvectors).

4. Matrices with no real eigenvalues, such as a $90 ^ \circ$ rotation. In this case there are no real eigenvectors either. However, if we are willing to use complex numbers then these matrices are like the previous case; they do have complex eigenvalues with corresponding eigenvectors.

5. Matrices that have only one eigenvalue, other than scalar matrices. Examples are (i) shears, such as the matrix $\matp{rr}{1&1\\ 0&1}$; (ii) nonzero matrices that are ``nilpotent'', which means some power is the zero matrix, for example $\matp{rr}{0&1\\ 0&0}$; and (iii) nilpotent matrices added to scalar matrices, as in $\matp{rr}{3&1\\ 0&3}$. These are strange in many ways. For example, they have only one line of eigenvectors.

   Non-scalar $2 \times 2$ matrices with only one eigenvalue are called ``defective''.

You can begin to see how any $2 \times 2$ matrix must be like one of these examples:

1. Find the characteristic polynomial, which has degree 2.

2. Factor it to get the roots (eigenvalues). You may need to use the quadratic formula, and the roots might be complex.

3. If the roots are real and distinct (different), then we get the two lines of eigenvectors in $\mathbb{R}^ 2$.

4. If the roots are are identical (for example, if the characteristic polynomial is $(\lambda-3) ^ 2$), then the matrix is either scalar or else is case .

5. Finally, if the roots are complex rather than real (in which case they are distinct), the matrix fits case .

All this is assuming that the matrix has real entries, but these cases are pretty much the same even for matrices with complex entries.

It turns out that even $n \times n$ matrices split into pieces that fit these cases.

Problem U-10. Classify the eight examples in the handout with pictures of a house according to this scheme. (The first example is the identity matrix.)

Problem U-11. Show that a $2 \times 2$ matrix is diagonalizable if and only if it is not defective. [This problem really should go after the next section. Also, ``diagonalizable'' here means possibly using complex numbers.]